The moduli spaces \(\mathbf {M}_v\)  and the fibers \(M_v\)  depend on the parity of the integer \(\chi \)  in the Mukai vector \(v = (0,2\theta ,\chi )\) . If \(\chi \)  is odd, then the Mukai vector \(v\)  is primitive and \(\mathbf {M}_v\)  is smooth of dimension 10 and the fiber \(M_v\)  is deformation equivalent to a generalized Kummer variety of dimension 6 (see [1]}). If \(\chi \)  is even, then the Mukai vector \(v = 2v^{\prime }\)  is twice a primitive Mukai vector and Rapagnetta shows in [2]} that the moduli space \(\mathbf {M}_v\)  and the fiber \(M_v\)  are both reduced, but singular. The singular locus \(\Sigma _v \subset M_v\)  parameterizes polystable sheaves of the form \({F}_1 \oplus {F}_2\)  with \(v({F}_i) = v^{\prime }\) , determinant bundle isomorphic to \(\mathcal {O}_J(2\theta )\)  and second Chern class summing up to 0. The locus \(\Sigma _v\)  is itself singular and its singular locus, \(\Omega _v \subset \Sigma _v\) , parameterizes polystable sheaves of the form \({F}\oplus {F}\) . The fiber \(M_v\)  admits a symplectic resolution \(\pi _v \colon \widetilde{M}_v \rightarrow M_v\) , where \(\widetilde{M}_v\)  is an irreducible holomorphic symplectic manifold which is birational, and hence deformation equivalent, to O'Grady's six dimensional exceptional example (see [2]}).
